What do the parameters of the Cycles Noise Texture do?
I don't think having a math formula will help much in getting the visual effect you want. But if you are really into understanding the math implemented in blender, I think your best chance is looking into the code. The relevant files are /intern/cycles/kernel/shaders/node_noise_texture.osl and /intern/cycles/kernel/shaders/node_texture.h.

I am not familiar with Perlin noise, but looking into the code, "detail" seems to refer to the number of octaves. It is hard for me to understand what "distortion" does, but noise is generated not at original point, but at a point shifted by random (and strange) amount proportional to the distortion.

Since you are more familiar with the stuff I think you'll be able to get more out of it (and maybe post it as an answer).